Get["testPackage`"] (* Doesn't appear to do anything or get any response from Mma *)
Get[FileNameJoin[{$UserBaseDirectory, "Applications", "testPackage`"}]]
$Failed
Needs["testPackage`"]
Needs::nocont: Context testPackage` was not created when Needs was evaluated. >>

Nothing works.

You can see the package installed in the ...Mathematica/Applications directory:

Also, you installed into $UserBaseDirectory, i.e. choose "Install for this user only", correct? I repeated the exact steps you describe (as far as I can tell), and it works correctly here. The question is what we did differently.
–
SzabolcsJan 28 '13 at 22:35

Do not put your code directly in the $UserBaseDirectory/Applications folder. Rather, create a new folder in the Applications folder, MyTestApplication say. Then write your BeginPackage statement as:

BeginPackage["MyTestApplication`testPackage`"]

Load the package with:

<< MyTestApplication`testPackage`

Or, create a Kernel folder in the MyTestApplication folder and create an init.m file in the Kernel folder that contains the above Get statement. Then you can load the package with:

<< MyTestApplication`

Mathematica will execute the init.m file and load the package. You could even have multiple packages and have the init.m file load each of them.

Users seem to constantly stumble over where to put packages. Create an application within the Applications folder and put the package THERE. That's where Mathematica looks for them.

Adding a little more discussion:

It would help if users thought a little less about "Packages" and a little more about "Applications". An application is a folder where you can collect your work on some topic: a book you are studying, or a course you are taking, or a research and development project. Where do you put your application folder? In the Mathematica$UserBaseDirector/Applications folder - where else? Your application folder might or might not contain package files. It may contain many notebooks. It may contain a whole folder structure. If you write packages, which is fairly easy to do, you could also add documentation through Workbench. It is a great way to organize, preserve, document and communicate your work. It is the fruit of your labor.

So, most of all, create those applications folders for your various projects.

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